This expression amounts to the leading contribution of the renowned (one-loop) Heisenberg-Euler effective Lagrangian [1]}. It gives rise to the effective coupling of four electromagnetic fields mediated by quantum vacuum fluctuations of electrons and positrons.
Higher loop corrections are parametrically suppressed with powers of \(\alpha =e^2/(4\pi )\simeq 1/137\) ; see Ref. [2]} for the coefficient in Eq. (REF ) at two loops. Hence, given that the above conditions (REF ) and (REF ) are met, Eq. (REF ) should allow for the accurate theoretical study of light propagation phenomena in the presence of experimentally realistic macroscopic electromagnetic fields. However, particularly due to the non-convergent nature of the employed expansions (cf. Sec. REF ), sizable deviations are to be expected when any of the parameters in Eqs. (REF ) and (REF ) becomes of the order of \(\lesssim 1\) .
